# Properties

 Label 2-675-1.1-c3-0-73 Degree $2$ Conductor $675$ Sign $-1$ Analytic cond. $39.8262$ Root an. cond. $6.31080$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.45·2-s + 11.8·4-s − 5.08·7-s + 17.3·8-s − 58.3·11-s − 21.2·13-s − 22.6·14-s − 17.8·16-s − 68.8·17-s − 40.8·19-s − 259.·22-s − 144.·23-s − 94.5·26-s − 60.3·28-s + 220.·29-s + 291.·31-s − 218.·32-s − 307.·34-s − 260.·37-s − 182.·38-s + 169.·41-s + 438.·43-s − 692.·44-s − 643.·46-s − 255.·47-s − 317.·49-s − 252.·52-s + ⋯
 L(s)  = 1 + 1.57·2-s + 1.48·4-s − 0.274·7-s + 0.765·8-s − 1.59·11-s − 0.452·13-s − 0.432·14-s − 0.278·16-s − 0.982·17-s − 0.492·19-s − 2.51·22-s − 1.30·23-s − 0.713·26-s − 0.407·28-s + 1.40·29-s + 1.68·31-s − 1.20·32-s − 1.54·34-s − 1.15·37-s − 0.776·38-s + 0.646·41-s + 1.55·43-s − 2.37·44-s − 2.06·46-s − 0.792·47-s − 0.924·49-s − 0.672·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$675$$    =    $$3^{3} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$39.8262$$ Root analytic conductor: $$6.31080$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{675} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 675,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 - 4.45T + 8T^{2}$$
7 $$1 + 5.08T + 343T^{2}$$
11 $$1 + 58.3T + 1.33e3T^{2}$$
13 $$1 + 21.2T + 2.19e3T^{2}$$
17 $$1 + 68.8T + 4.91e3T^{2}$$
19 $$1 + 40.8T + 6.85e3T^{2}$$
23 $$1 + 144.T + 1.21e4T^{2}$$
29 $$1 - 220.T + 2.43e4T^{2}$$
31 $$1 - 291.T + 2.97e4T^{2}$$
37 $$1 + 260.T + 5.06e4T^{2}$$
41 $$1 - 169.T + 6.89e4T^{2}$$
43 $$1 - 438.T + 7.95e4T^{2}$$
47 $$1 + 255.T + 1.03e5T^{2}$$
53 $$1 - 214.T + 1.48e5T^{2}$$
59 $$1 + 331.T + 2.05e5T^{2}$$
61 $$1 - 54.9T + 2.26e5T^{2}$$
67 $$1 + 758.T + 3.00e5T^{2}$$
71 $$1 - 904.T + 3.57e5T^{2}$$
73 $$1 + 866.T + 3.89e5T^{2}$$
79 $$1 - 206.T + 4.93e5T^{2}$$
83 $$1 + 463.T + 5.71e5T^{2}$$
89 $$1 + 601.T + 7.04e5T^{2}$$
97 $$1 + 229.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$